Geometry and topology in dimension 3
京都大学 数理解析研究所 202号室
Professor John Morgan (Columbia University)
The Poincare Conjecture, that every simply connected 3-manifold is homeomorphic to the sphere, has been a central question in topology since its formulation in 1904. In the early 1980s Thurston gave a vast generalization and reformulation of this question that goes under the name of the `Geometrization Conjecture' It differed in two essential ways: (i) it covered all 3-manifolds, not just simply connected ones, and (ii) it was not a purely topological conjecture; it mixed geometric and topological notions. This conjecture suggested that every 3-manifold could be cut in a canonical way so that the resulting pieces all have especially nice Riemannian metrics, metrics derived from Lie groups. It easily implies the Poincare Conjecture. Richard Hamilton suggested a method to attack the problem of constructing the metrics required by Thurston's conjecture. His method was to begin with any Riemannian metric on the 3-manifold under study and deform this metric by a non-linear parabolic equation (a flow equation) which is a non-linear heat-type equation. It is called the Ricci flow equation. The intuition is that just as heat flow distributes the temperature evenly, this equation should distribute the curvature evenly over the manifold and hence produce the homogeneous metrics that the Geometrization Conjecture posits. There is an essential difficulty in this approach coming from the fact that the evolution equation is non-linear. This means that singularities can develop in finite-time stopping the evolution before it reaches its homogeneous limit at time = +infinty. In 2002/2003 Perelman posed a series of preprints explaining how to deal with these finite-time singularities and continue the flow past them, constructing what is known as a Ricci flow with surgery. He went on to justify Hamilton's vision and complete the proof of the Geometrization Conjecture. In this talk I will discuss the various types of geometric 3-manifolds, state Thurston's conjecture, give an overview of the work of Hamilton, and finally a brief introduction to Perelman's proof.